*Center for Computer Research in Music
and Acoustics (CCRMA)*

**December 19, 1997**

Stanford University,
Stanford, CA 94305 USA

Davide Rocchesso

Centro di Sonologia Computazionale

Dipartimento di Elettronica e Informatica

Università degli Studi di Padova

via Gradenigo, 6/A - 35131 Padova, ITALY

This paper collects together various facts about digital waveguide
networks (DWN) used in acoustic modeling, particularly results
pertaining to lossless scattering at the junction of intersecting
digital waveguides. Applications discussed include music synthesis
based on physical models and delay effects such as artificial
reverberation. Connections with Wave Digital Filters (WDF),
ladder/lattice digital filters, and other related topics are outlined.
General conditions for losslessness and passivity are specified.
Computational complexity and dynamic range requirements are addressed.
Both physical and algebraic analyses are utilized. The physical
interpretation leads to many of the desirable properties of DWNs.
Using both physical and algebraic approaches, three new normalized
ladder filter structures are derived which have only three
multiplications per two-port scattering junction instead of the four
required in the well known version. A vector scattering formulation
is derived which maximizes generality subject to maintaining desirable
properties. Scattering junctions are generalized to allow any
waveguide to have a complex wave impedance which is equivalent at
the junction to a lumped load impedance, thus providing a convenient
bridge between lumped and distributed modeling methods. Junctions
involving complex wave impedances yield generalized scattering
coefficients which are frequency dependent and therefore implemented
in practice using digital filters. Scattering filters are typically
isolable to one per junction in a manner analogous to the multiply in
a one-multiply lattice-filter section.

**Detailed Contents (and Navigation)**

- Introduction
- Digital Filters as Physical Models
- Digital Waveguide Networks (DWN)
- Properties of DWNs
- Paper Outline

- Basic DWN Formulation
- The Ideal Acoustic Tube
- Multivariable Formulation of the Acoustic Tube
- Generalized Wave Impedance
- Generalized Complex Signal Power
- Medium Passivity
- Impulsive Signals Interpretation
- Bandlimited Signals Interpretation
- Interpolated Digital Waveguides
- Lossy, Dispersive Waveguides
- The General Linear Time-Invariant Case

- The Lossless Junction

- The Normalized Lossless Junction
- Time-Varying Normalized Waveguide Networks
- Isolating Time-Varying Junctions
- Scattering of Normalized Waves

- Physical Scattering Junctions
- Parallel Junction of Multivariable Complex Waveguides
- Loaded Junctions
- Example: Bridge Coupling of Piano Strings

- Nonlinear, Time-Varying DWNs
- Algebraic Properties of Lossless Junctions

- Complexity Reduction: A Physical Approach

- Complexity Reduction: A Geometric Approach

- Non-constant scattering matrices and applications

- Conclusions
- Appendix A: The Digital Waveguide Transformer
- Bibliography
- About this document ...

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